Suppose I have an inequality between 2 functions in a certain domain $[a,b]$ e.g. $f(x)\leq g(x)$ for all $x\in [a,b]$. And I have a random variable $X$ that is bounded by the same domain. In addition, there's an event $A$ on which I want to condition $X$. Is it correct to say that $\mathbb{E}[f(X)|A] \leq \mathbb{E}[g(X)|A]$?
You can also assume $\Pr[A]>0$.